ODE
\[ -y(x) \left (m^2-n (n+1) \left (1-x^2\right )\right )+\left (1-x^2\right )^2 y''(x)-2 x \left (1-x^2\right ) y'(x)=0 \] ODE Classification
[[_2nd_order, _with_linear_symmetries]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.172357 (sec), leaf count = 20
\[\{\{y(x)\to c_1 P_n^m(x)+c_2 Q_n^m(x)\}\}\]
Maple ✓
cpu = 0.31 (sec), leaf count = 17
\[[y \left (x \right ) = \textit {\_C1} \LegendreP \left (n , m , x\right )+\textit {\_C2} \LegendreQ \left (n , m , x\right )]\] Mathematica raw input
DSolve[-((m^2 - n*(1 + n)*(1 - x^2))*y[x]) - 2*x*(1 - x^2)*y'[x] + (1 - x^2)^2*y''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> C[1]*LegendreP[n, m, x] + C[2]*LegendreQ[n, m, x]}}
Maple raw input
dsolve((-x^2+1)^2*diff(diff(y(x),x),x)-2*x*(-x^2+1)*diff(y(x),x)-(m^2-n*(n+1)*(-x^2+1))*y(x) = 0, y(x))
Maple raw output
[y(x) = _C1*LegendreP(n,m,x)+_C2*LegendreQ(n,m,x)]